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The pp-Bures–Wasserstein distance via 3×33\times3 block matrices (with application to pp-Bures–Wasserstein barycenters)

A mix of human-written and AI-generated textHuman understanding: all partsmath.FA — Functional Analysismath.MG — Metric Geometrymath.OC — Optimization and Control

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026/ arXiv: 2610.15889v1 / CC BY 4.0

Abstract

We study the pp-Bures–Wasserstein distance Dp(A,B)=(tr Ap+tr Bp−2∥A1/2B1/2∥pp)1/2D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2} on positive semidefinite matrices; it equals the Bures–Wasserstein distance for p=1p=1, and the Hilbert–Schmidt distance for p=2p=2. We prove that DpD_p is a metric for every 1≤p≤21\le p\le2, by proving a stronger 2014 conjecture of Lin and van den Driessche on 3×33\times3 block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for DpD_p by a fixed-point equation and prove geometric convergence of the associated iteration.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The AI-driven side of this work began in October 2025; the first complete proof was obtained on 11 August 2026.

Tools used

OpenAI
GPT Pro

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026